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Добірка наукової літератури з теми "Finite semigroups"

Автор: Grafiati

Опубліковано: 7 грудня 2024

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Статті в журналах з теми "Finite semigroups"

LE SAEC, BERTRAND, JEAN-ERIC PIN, and PASCAL WEIL. "SEMIGROUPS WITH IDEMPOTENT STABILIZERS AND APPLICATIONS TO AUTOMATA THEORY." International Journal of Algebra and Computation 01, no. 03 (September 1991): 291–314. http://dx.doi.org/10.1142/s0218196791000195.

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Nous prouvons que tout semigroupe fini est quotient d'un semigroupe fini dans lequel les stabilisateurs droits satisfont les identités x = x2 et xy = xyx. Ce resultat a plusieurs consé-quences. Tout d'abord, nous l'utilisons, en même temps qu'un résultat de I. Simon sur les congruences de chemins, pour obtenir une preuve purement algébrique d'un théorème profond de McNaughton sur les mots infinis. Puis, nous donnons une preuve algébrique d'un théorème de Brown sur des conditions de finitude pour les semigroupes. We show that every finite semigroup is a quotient of a finite semigroup in which every right stabilizer satisfies the identities x = x2 and xy = xyx. This result has several consequences. We first use it together with a result of I. Simon on congruences on paths to obtain a purely algebraic proof of a deep theorem of McNaughton on infinite words. Next, we give an algebraic proof of a theorem of Brown on a finiteness condition for semigroups.

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Shoji, Kunitaka. "Regular Semigroups Which Are Amalgamation Bases for Finite Semigroups." Algebra Colloquium 14, no. 02 (June 2007): 245–54. http://dx.doi.org/10.1142/s1005386707000247.

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In this paper, we prove that a completely 0-simple (or completely simple) semigroup is an amalgamation base for finite semigroups if and only if it is an amalgamation base for semigroups. By adopting the same method as used in a previous paper, we prove that a finite regular semigroup is an amalgamation base for finite semigroups if its [Formula: see text]-classes are linearly ordered and all of its principal factor semigroups are amalgamation bases for finite semigroups. Finally, we give an example of a finite semigroup U which is an amalgamation base for semigroups, but not all of its principal factor semigroups are amalgamation bases either for semigroups or for finite semigroups.

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Guo, Xiaojiang, and Lin Chen. "Semigroup algebras of finite ample semigroups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 142, no. 2 (March 21, 2012): 371–89. http://dx.doi.org/10.1017/s0308210510000715.

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An adequate semigroup S is called ample if ea = a(ea)* and ae = (ae)†a for all a ∈ S and e ∈ E(S). Inverse semigroups are exactly those ample semigroups that are regular. After obtaining some characterizations of finite ample semigroups, it is proved that semigroup algebras of finite ample semigroups have generalized triangular matrix representations. As applications, the structure of the radicals of semigroup algebras of finite ample semigroups is obtained. In particular, it is determined when semigroup algebras of finite ample semigroup are semiprimitive.

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Birget, Jean-Camille, Stuart Margolis, and John Rhodes. "Semigroups whose idempotents form a subsemigroup." Bulletin of the Australian Mathematical Society 41, no. 2 (April 1990): 161–84. http://dx.doi.org/10.1017/s0004972700017986.

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We prove that if the “type-II-construct” subsemigroup of a finite semigroup S is regular, then the “type-II” subsemigroup of S is computable (actually in this case, type-II and type-II-construct are equal). This, together with certain older results about pseudo-varieties of finite semigroups, leads to further results:(1) We get a new proof of Ash's theorem: If the idempotents in a finite semigroup S commute, then S divides a finite inverse semigroup. Equivalently: The pseudo-variety generated by the finite inverse semigroups consists of those finite semigroups whose idempotents commute.(2) We prove: If the idempotents of a finite semigroup S form a subsemigroup then S divides a finite orthodox semigroup. Equivalently: The pseudo-variety generated by the finite orthodox semigroups consists of those finite semigroups whose idempotents form a subsemigroup.(3) We prove: The union of all the subgroups of a semigroup S forms a subsemigroup if and only if 5 belongs to the pseudo-variety u * G if and only if Sn belongs to u. Here u denotes the pseudo-variety of finite semigroups which are unions of groups.For these three classes of semigroups, type-II is equal to type-II construct.

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VERNITSKI, ALEXEI. "ORDERED AND $\mathcal{J}$-TRIVIAL SEMIGROUPS AS DIVISORS OF SEMIGROUPS OF LANGUAGES." International Journal of Algebra and Computation 18, no. 07 (November 2008): 1223–29. http://dx.doi.org/10.1142/s021819670800486x.

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A semigroup of languages is a set of languages considered with (and closed under) the operation of catenation. In other words, semigroups of languages are subsemigroups of power-semigroups of free semigroups. We prove that a (finite) semigroup is positively ordered if and only if it is a homomorphic image, under an order-preserving homomorphism, of a (finite) semigroup of languages. Hence it follows that a finite semigroup is [Formula: see text]-trivial if and only if it is a homomorphic image of a finite semigroup of languages.

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Almeida, J., M. H. Shahzamanian, and M. Kufleitner. "Nilpotency and strong nilpotency for finite semigroups." Quarterly Journal of Mathematics 70, no. 2 (November 21, 2018): 619–48. http://dx.doi.org/10.1093/qmath/hay059.

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AbstractNilpotent semigroups in the sense of Mal’cev are defined by semigroup identities. Finite nilpotent semigroups constitute a pseudovariety, MN, which has finite rank. The semigroup identities that define nilpotent semigroups lead us to define strongly Mal’cev nilpotent semigroups. Finite strongly Mal’cev nilpotent semigroups constitute a non-finite rank pseudovariety, SMN. The pseudovariety SMN is strictly contained in the pseudovariety MN, but all finite nilpotent groups are in SMN. We show that the pseudovariety MN is the intersection of the pseudovariety BGnil with a pseudovariety defined by a κ-identity. We further compare the pseudovarieties MN and SMN with the Mal’cev product 𝖩ⓜ𝖦nill.

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Ash, C. J., and T. E. Hall. "Finite semigroups with commuting idempotents." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 43, no. 1 (August 1987): 81–90. http://dx.doi.org/10.1017/s1446788700028998.

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AbstractWe show that every such semigroup is a homomorphic image of a subsemigroup of some finite inverse semigroup. This shows that the pseudovariety generated by the finite inverse semigroups consists of exactly the finite semigroups with commuting idempotents.

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IWAKI, E., E. JESPERS, S. O. JURIAANS, and A. C. SOUZA FILHO. "HYPERBOLICITY OF SEMIGROUP ALGEBRAS II." Journal of Algebra and Its Applications 09, no. 06 (December 2010): 871–76. http://dx.doi.org/10.1142/s0219498810004270.

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In 1996, Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki–Juriaans–Souza Filho continued this line of research by partially classifying the finite semigroups whose rational semigroup algebra contains a ℤ-order with hyperbolic unit group. In this paper, we complete this classification and give an easy proof that deals with all finite semigroups.

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JACKSON, DAVID A. "DECISION AND SEPARABILITY PROBLEMS FOR BAUMSLAG–SOLITAR SEMIGROUPS." International Journal of Algebra and Computation 12, no. 01n02 (February 2002): 33–49. http://dx.doi.org/10.1142/s0218196702000857.

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We show that the semigroups Sk,ℓ having semigroup presentations <a, b:abk = bℓ a> are residually finite and finitely separable. Generally, these semigroups have finite separating images which are finite groups and other finite separating images which are semigroups of order-increasing transformations on a finite partially ordered set. These semigroups thus have vastly different residual and separability properties than the Baumslag–Solitar groups which contain them.

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Dolinka, Igor, and Robert D. Gray. "Universal locally finite maximally homogeneous semigroups and inverse semigroups." Forum Mathematicum 30, no. 4 (July 1, 2018): 947–71. http://dx.doi.org/10.1515/forum-2017-0074.

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Abstract In 1959, Philip Hall introduced the locally finite group {\mathcal{U}} , today known as Hall’s universal group. This group is countable, universal, simple, and any two finite isomorphic subgroups are conjugate in {\mathcal{U}} . It can explicitly be described as a direct limit of finite symmetric groups. It is homogeneous in the model-theoretic sense since it is the Fraïssé limit of the class of all finite groups. Since its introduction Hall’s group and several natural generalisations have been studied widely. In this article we use a generalisation of Fraïssé’s theory to construct a countable, universal, locally finite semigroup {\mathcal{T}} , that arises as a direct limit of finite full transformation semigroups, and has the highest possible degree of homogeneity. We prove that it is unique up to isomorphism among semigroups satisfying these properties. We prove an analogous result for inverse semigroups, constructing a maximally homogeneous universal locally finite inverse semigroup {\mathcal{I}} which is a direct limit of finite symmetric inverse semigroups (semigroups of partial bijections). The semigroups {\mathcal{T}} and {\mathcal{I}} are the natural counterparts of Hall’s universal group for semigroups and inverse semigroups, respectively. While these semigroups are not homogeneous, they still exhibit a great deal of symmetry. We study the structural features of these semigroups and locate several well-known homogeneous structures within them, such as the countable generic semilattice, the countable random bipartite graph, and Hall’s group itself.

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Дисертації з теми "Finite semigroups"

Wilson, Wilf A. "Computational techniques in finite semigroup theory." Thesis, University of St Andrews, 2019. http://hdl.handle.net/10023/16521.

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A semigroup is simply a set with an associative binary operation; computational semigroup theory is the branch of mathematics concerned with developing techniques for computing with semigroups, as well as investigating semigroups with the help of computers. This thesis explores both sides of computational semigroup theory, across several topics, especially in the finite case. The central focus of this thesis is computing and describing maximal subsemigroups of finite semigroups. A maximal subsemigroup of a semigroup is a proper subsemigroup that is contained in no other proper subsemigroup. We present novel and useful algorithms for computing the maximal subsemigroups of an arbitrary finite semigroup, building on the paper of Graham, Graham, and Rhodes from 1968. In certain cases, the algorithms reduce to computing maximal subgroups of finite groups, and analysing graphs that capture information about the regular I-classes of a semigroup. We use the framework underpinning these algorithms to describe the maximal subsemigroups of many families of finite transformation and diagram monoids. This reproduces and greatly extends a large amount of existing work in the literature, and allows us to easily see the common features between these maximal subsemigroups. This thesis is also concerned with direct products of semigroups, and with a special class of semigroups known as Rees 0-matrix semigroups. We extend known results concerning the generating sets of direct products of semigroups; in doing so, we propose techniques for computing relatively small generating sets for certain kinds of direct products. Additionally, we characterise several features of Rees 0-matrix semigroups in terms of their underlying semigroups and matrices, such as their Green's relations and generating sets, and whether they are inverse. In doing so, we suggest new methods for computing Rees 0-matrix semigroups.

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Distler, Andreas. "Classification and enumeration of finite semigroups." Thesis, St Andrews, 2010. http://hdl.handle.net/10023/945.

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Hum, Marcus. "The representation theory of finite semigroups /." Thesis, McGill University, 2000. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=33409.

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Rodgers, James David, and jdr@cgs vic edu au. "On E-Pseudovarieties of Finite Regular Semigroups." RMIT University. Mathematical and Geospatial Sciences, 2007. http://adt.lib.rmit.edu.au/adt/public/adt-VIT20080808.155720.

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An e-pseudovariety is a class of finite regular semigroups closed under the taking of homomorphic images, regular subsemigroups and finite direct products. Chapter One consists of a survey of those results from algebraic semigroup theory, universal algebra and lattice theory which are used in the following two chapters. In Chapter Two, a theory of generalised existence varieties is developed. A generalised existence variety is a class of regular semigroups closed under the taking of homomorphic images, regular subsemigroups, finite direct products and arbitrary powers. Equivalently, a generalised e-variety is the union of a directed family of existence varieties. It is demonstrated that a class of finite regular semigroups is an e-pseudovariety if and only if the class consists only of the finite members of some generalised existence variety. The relationship between certain lattices of e-pseudovarieties and generalised existence varieties is explored and a usefu l complete surjective lattice homomorphism is found. A study of complete congruences on lattices of existence varieties and e-pseudovarieties forms Chapter Three. In particular it is shown that a certain meet congruence, whose description is relatively simple, can be extended to yield a complete congruence on a lattice of e-pseudovarieties of finite regular semigroups. Ultimately, theorems describing the method of construction of all complete congruences of lattices of e-pseudovarieties whose members are finite E-solid or locally inverse regular semigroups are proved.

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Distler, Andreas [Verfasser]. "Classification and Enumeration of Finite Semigroups / Andreas Distler." Aachen : Shaker, 2010. http://d-nb.info/1081886196/34.

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Tesson, Pascal. "Computational complexity questions related to finite monoids and semigroups." Thesis, McGill University, 2003. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=84441.

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In this thesis, we address a number of issues pertaining to the computational power of monoids and semigroups as machines and to the computational complexity of problems whose difficulty is parametrized by an underlying semigroup or monoid and find that these two axes of research are deeply intertwined.
We first consider the "program over monoid" model of D. Barrington and D. Therien [BT88] and set out to answer two fundamental questions: which monoids are rich enough to recognize arbitrary languages via programs of arbitrary length, and which monoids are so weak that any program over them has an equivalent of polynomial length? We find evidence that the two notions are dual and in particular prove that every monoid in DS has exactly one of these two properties. We also prove that for certain "weak" varieties of monoids, programs can only recognize those languages with a "neutral letter" that can be recognized via morphisms over that variety.
We then build an algebraic approach to communication complexity, a field which has been of great importance in the study of small complexity classes. We prove that every monoid has communication complexity O(1), &THgr;(log n) or &THgr;(n) in this model. We obtain similar classifications for the communication complexity of finite monoids in the probabilistic, simultaneous, probabilistic simultaneous and MOD p-counting variants of this two-party model and thus characterize the communication complexity (in a worst-case partition sense) of every regular language in these five models. Furthermore, we study the same questions in the Chandra-Furst-Lipton multiparty extension of the classical communication model and describe the variety of monoids which have bounded 3-party communication complexity and bounded k-party communication complexity for some k. We also show how these bounds can be used to establish computational limitations of programs over certain classes of monoids.
Finally, we consider the computational complexity of testing if an equation or a system of equations over some fixed finite monoid (or semigroup) has a solution.

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Garba, Goje Uba. "Idempotents, nilpotents, rank and order in finite transformation semigroups." Thesis, University of St Andrews, 1992. http://hdl.handle.net/10023/13703.

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AlAli, Amal. "Cosets in inverse semigroups and inverse subsemigroups of finite index." Thesis, Heriot-Watt University, 2016. http://hdl.handle.net/10399/3185.

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The index of a subgroup of a group counts the number of cosets of that subgroup. A subgroup of finite index often shares structural properties with the group, and the existence of a subgroup of finite index with some particular property can therefore imply useful structural information for the overgroup. Although a developed theory of cosets in inverse semigroups exists, it is defined only for closed inverse subsemigroups, and the structural correspondences between an inverse semigroup and a closed inverse subsemigroup of finte index are much weaker than in the group case. Nevertheless, many aspects of this theory remain of interest, and some of them are addressed in this thesis. We study the basic theory of cosets in inverse semigroups, including an index formula for chains of subgroups and an analogue of M. Hall’s Theorem on counting subgroups of finite index in finitely generated groups. We then look at specific examples, classifying the finite index inverse subsemigroups in polycyclic monoids and in graph inverse semigroups. Finally, we look at the connection between the properties of finite generation and having finte index: these were shown to be equivalent for free inverse monoids by Margolis and Meakin.

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Abu-Ghazalh, Nabilah Hani. "Finiteness conditions for unions of semigroups." Thesis, University of St Andrews, 2013. http://hdl.handle.net/10023/3687.

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In this thesis we prove the following: The semigroup which is a disjoint union of two or three copies of a group is a Clifford semigroup, Rees matrix semigroup or a combination between a Rees matrix semigroup and a group. Furthermore, the semigroup which is a disjoint union of finitely many copies of a finitely presented (residually finite) group is finitely presented (residually finite) semigroup. The constructions of the semigroup which is a disjoint union of two copies of the free monogenic semigroup are parallel to the constructions of the semigroup which is a disjoint union of two copies of a group, i.e. such a semigroup is Clifford (strong semilattice of groups) or Rees matrix semigroup. However, the semigroup which is a disjoint union of three copies of the free monogenic semigroup is not just a strong semillatice of semigroups, Rees matrix semigroup or combination between a Rees matrix semigroup and a semigroup, but there are two more semigroups which do not arise from the constructions of the semigroup which is a disjoint union of three copies of a group. We also classify semigroups which are disjoint unions of two or three copies of the free monogenic semigroup. There are three types of semigroups which are unions of two copies of the free monogenic semigroup and nine types of semigroups which are unions of three copies of the free monogenic semigroup. For each type of such semigroups we exhibit a presentation defining semigroups of this type. The semigroup which is a disjoint union of finitely many copies of the free monogenic semigroup is finitely presented, residually finite, hopfian, has soluble word problem and has soluble subsemigroup membership problem.

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Awang, Jennifer S. "Dots and lines : geometric semigroup theory and finite presentability." Thesis, University of St Andrews, 2015. http://hdl.handle.net/10023/6923.

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Geometric semigroup theory means different things to different people, but it is agreed that it involves associating a geometric structure to a semigroup and deducing properties of the semigroup based on that structure. One such property is finite presentability. In geometric group theory, the geometric structure of choice is the Cayley graph of the group. It is known that in group theory finite presentability is an invariant under quasi-isometry of Cayley graphs. We choose to associate a metric space to a semigroup based on a Cayley graph of that semigroup. This metric space is constructed by removing directions, multiple edges and loops from the Cayley graph. We call this a skeleton of the semigroup. We show that finite presentability of certain types of direct products, completely (0-)simple, and Clifford semigroups is preserved under isomorphism of skeletons. A major tool employed in this is the Švarc-Milnor Lemma. We present an example that shows that in general, finite presentability is not an invariant property under isomorphism of skeletons of semigroups, and in fact is not an invariant property under quasi-isometry of Cayley graphs for semigroups. We give several skeletons and describe fully the semigroups that can be associated to these.

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Книги з теми "Finite semigroups"

Ganyushkin, Olexandr, and Volodymyr Mazorchuk. Classical Finite Transformation Semigroups. London: Springer London, 2009. http://dx.doi.org/10.1007/978-1-84800-281-4.

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Rhodes, John, and Benjamin Steinberg. The q-theory of Finite Semigroups. Boston, MA: Springer US, 2009. http://dx.doi.org/10.1007/b104443.

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Volodymyr, Mazorchuk, ed. Classical finite transformation semigroups: An introduction. London: Springer, 2009.

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Koli︠a︡da, S. F. Dynamics and numbers: A special program, June 1-July 31, 2014, Max Planck Institute for Mathematics, Bonn, Germany : international conference, July 21-25, 2014, Max Planck Institute for Mathematics, Bonn, Germany. Edited by Max-Planck-Institut für Mathematik. Providence, Rhode Island: American Mathematical Society, 2016.

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Finite Semigroups and Universal Algebra. World Scientific Publishing Co Pte Ltd, 1995.

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Finite semigroups and universal algebra. Singapore: World Scientific, 1994.

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Finite Semigroups and Universal Algebra. World Scientific Publishing Co Pte Ltd, 1995.

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Steinberg, Benjamin. Representation Theory of Finite Monoids. Springer International Publishing AG, 2016.

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Steinberg, Benjamin. Representation Theory of Finite Monoids. Springer International Publishing AG, 2016.

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Rhodes, John, and Benjamin Steinberg. The q-theory of Finite Semigroups. Springer, 2010.

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Частини книг з теми "Finite semigroups"

Straubing, Howard. "Finite Semigroups." In Finite Automata, Formal Logic, and Circuit Complexity, 53–78. Boston, MA: Birkhäuser Boston, 1994. http://dx.doi.org/10.1007/978-1-4612-0289-9_5.

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Ash, C. J. "Finite Idempotent-Commuting Semigroups." In Semigroups and Their Applications, 13–23. Dordrecht: Springer Netherlands, 1987. http://dx.doi.org/10.1007/978-94-009-3839-7_2.

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Pin, J. E. "Structure of Finite Semigroups." In Varieties of Formal Languages, 45–78. Boston, MA: Springer US, 1986. http://dx.doi.org/10.1007/978-1-4613-2215-3_4.

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Renner, Lex E. "Finite Reductive Monoids." In Semigroups, Formal Languages and Groups, 369–80. Dordrecht: Springer Netherlands, 1995. http://dx.doi.org/10.1007/978-94-011-0149-3_12.

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Gil’, Michael I. "Strongly Continuous Semigroups." In Stability of Finite and Infinite Dimensional Systems, 261–84. Boston, MA: Springer US, 1998. http://dx.doi.org/10.1007/978-1-4615-5575-9_13.

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Pin, J. E. "Power Semigroups and Related Varieties of Finite Semigroups." In Semigroups and Their Applications, 139–52. Dordrecht: Springer Netherlands, 1987. http://dx.doi.org/10.1007/978-94-009-3839-7_18.

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Hall, T. E. "Finite Inverse Semigroups and Amalgamation." In Semigroups and Their Applications, 51–56. Dordrecht: Springer Netherlands, 1987. http://dx.doi.org/10.1007/978-94-009-3839-7_7.

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Froidure, Véronique, and Jean-Eric Pin. "Algorithms for computing finite semigroups." In Foundations of Computational Mathematics, 112–26. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/978-3-642-60539-0_9.

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Rhodes, John, and Benjamin Steinberg. "The Complexity of Finite Semigroups." In Springer Monographs in Mathematics, 1–172. Boston, MA: Springer US, 2008. http://dx.doi.org/10.1007/978-0-387-09781-7_4.

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Kublanovskii, S. I. "Algorithmic Problems for Finite Groups and Finite Semigroups." In Algorithmic Problems in Groups and Semigroups, 161–70. Boston, MA: Birkhäuser Boston, 2000. http://dx.doi.org/10.1007/978-1-4612-1388-8_9.

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Тези доповідей конференцій з теми "Finite semigroups"

ALMEIDA, J. "DYNAMICS OF FINITE SEMIGROUPS." In Semigroups, Algorithms, Automata and Languages. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812776884_0009.

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BULATOV, ANDREI, PETER JEAVONS, and MIKHAIL VOLKOV. "FINITE SEMIGROUPS IMPOSING TRACTABLE CONSTRAINTS." In Semigroups, Algorithms, Automata and Languages. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812776884_0011.

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TROTTER, PETER G. "DECIDABILITY PROBLEMS IN FINITE SEMIGROUPS." In Semigroups, Algorithms, Automata and Languages. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812776884_0022.

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Kozhukhov, Igor Borisovich, and Ksenia Anatolievna Kolesnikova. "Some conditions of finiteness on polygons over semigroups." In Academician O.B. Lupanov 14th International Scientific Seminar "Discrete Mathematics and Its Applications". Keldysh Institute of Applied Mathematics, 2022. http://dx.doi.org/10.20948/dms-2022-68.

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Анотація:

A polygon over a semigroup is an algebraic model machine. A finiteness condition in algebra is any condition which is satisfied by all finite algebras. The following finiteness conditions in acts over semigroups: Artinianity, Noetherian, Hopfian, Kohopfian, Cantorian, Cocantorian, the relationship between them is discussed. In addition, issues are discussed preserving or not preserving these properties with respect to the take operation direct product and coproduct.

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RIBES, LUIS. "PROFINITE GROUPS AND APPLICATIONS TO FINITE SEMIGROUPS." In Semigroups, Algorithms, Automata and Languages. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812776884_0008.

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STRAUBING, HOWARD. "FINITE SEMIGROUPS AND THE LOGICAL DESCRIPTION OF REGULAR LANGUAGES." In Semigroups, Algorithms, Automata and Languages. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812776884_0020.

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ALMEIDA, JORGE. "FINITE SEMIGROUPS: AN INTRODUCTION TO A UNIFIED THEORY OF PSEUDOVARIETIES." In Semigroups, Algorithms, Automata and Languages. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812776884_0001.

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VOLKOV, M. V. "THE FINITE BASIS PROBLEM FOR FINITE SEMIGROUPS: A SURVEY." In Proceedings of the International Conference. WORLD SCIENTIFIC, 2000. http://dx.doi.org/10.1142/9789812792310_0017.

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FERNANDES, VíTOR H. "PRESENTATIONS FOR SOME MONOIDS OF PARTIAL TRANSFORMATIONS ON A FINITE CHAIN: A SURVEY." In Semigroups, Algorithms, Automata and Languages. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812776884_0015.

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Denecke, K., and Y. Susanti. "Semigroups of n-ary Operations on Finite Sets." In The International Conference on Algebra 2010 - Advances in Algebraic Structures. WORLD SCIENTIFIC, 2011. http://dx.doi.org/10.1142/9789814366311_0011.

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